Mathematical Analysis - Topological Spaces, Metric Spaces

Topological Spaces, Metric Spaces

The motivation for studying mathematical analysis in the wider context of topological or metric spaces is threefold:

1. The same basic techniques have proved applicable to a wider class of problems (e.g., the study of function spaces).
2. A greater understanding of analysis in more abstract spaces frequently proves to be directly applicable to classical problems. For example, in Fourier analysis, functions are expressed in terms of a certain infinite sum of trigonometric functions. Thus Fourier analysis might be used to decompose a sound into a unique combination of pure tones of various pitches. The "weights", or coefficients, of the terms in the Fourier expansion of a function can be thought of as components of a vector in an infinite dimensional space known as a Hilbert space. Study of functions defined in this more general setting thus provides a convenient method of deriving results about the way functions vary in space as well as time or, in more mathematical terms, partial differential equations, where this technique is known as separation of variables.
3. The conditions needed to prove the particular result are stated more explicitly. The analyst then becomes more aware exactly what aspect of the assumption is needed to prove the theorem.